===================================================== Lab 3: Differential Equations and Data ===================================================== :Author: Tim Reluga and Elissa Schwartz :Date: Tuesday, June 14, 2011 Objectives ================================================ * Use linear least squares to determine growth and decay rates in linear models. * Use nonlinear least squares to estimate parameters for parameter estimation. Procedures ================================================ 1) Open a browser to http://www.math.psu.edu/treluga/yaleimmunology/lab3 This directory contains files and background used in this lab. Oscillatory time-series data and nonlinear least squares --------------------------------------------------------- We can start with something that's "simple". The directory OSC contains a time series for a chemical species X involved in a system of reactions with oscillatory dynamics. Find parameter values for the autocatalytic reactions 0 -> X 0 -> Y X -> 0 ( rate = 1 ) 2 X + Y -> 3 X ( rate = 1 ) that fit this time-series. You can download the COPASI file so you don't have to enter things from scratch. The data have been non-dimensionalized, so that two of the reaction rates are 1, so you need only fit the other two and the initial conditions. Treatment of HIV patients --------------------------------------------------------- 2) Directory "HIV1" contains data files for two patients treated for HIV around 1994. These are from the paper Wei1995.pdf. Use COPASI to fit an exponential decay model to these data (linear least squares would suffice, but fitting the decay model is more natural in COPASI). * How do the rates you calculate compare to those calculated in Wei'95? How can you get your estimates to match theirs? * What does your estimated decay rate tell us about the the immunological dynamics? 3) Directory "HIV2" contains data files for HIV treatment patient with more frequent sampling. Use COPASI and nonlinear least squares to fit the model of Perelson1996.pdf to these data. Note that you have to fit both the parameters AND the initial conditions for each patients. (the patient3.txt data is a little different from the first two, and can be ignored) +----------------------------------------------+ | dT/dt = lambda - k T V - d T | | dI/dt = (1-epsilon_{RT}) k T V - delta I | | dV/dt = (1-epsilon_{PI}) p I - c V | | dW/dt = epsilon_{PI} p I - c W | +----------------------------------------------+ * What does your estimated decay rate tell us about the the immunological dynamics? * Why can this fit tell us more than the fit in (2)? * Do these data and model convince you that the rate of viral dynamics turnover is very fast (on the order a few hours) in HIV?